The sinusoidal modelling of sound signals such as music and speech is a powerful tool for parameterizing sound sources. Once a sound has been parameterized, it can be synthesized for example, with a different pitch and duration.
A sampled short time signal xn on which a window wn is applied may be represented by a model {tilde over (x)}n, consisting of a sum of K sinusoids which are characterized by their frequency wk, phase φk and amplitude ak,
                                          x            ~                    n                =                              w            n                    ⁢                                    ∑                              k                =                0                                            K                -                1                                      ⁢                                          a                k                            ⁢                              cos                ⁡                                  (                                                            2                      ⁢                                                                                          ⁢                      π                      ⁢                                                                                          ⁢                                              ω                        k                                            ⁢                                                                        n                          -                                                      n                            0                                                                          N                                                              +                                          ϕ                      k                                                        )                                                                                        (        1        )            The offset value n0 allows the origin of the timescale to be placed exactly in the middle of the window. For a signal with length N, n0 equals
            N      -      1        2    .
If the signal would be synthesized by a bank of oscillators, the complexity would be O(NK) with N being the number of samples and K the number of sinusoidal components. As described in patent WO 93/03478, the computational efficiency of the synthesis can be improved by using an inverse fourier transform. However, the method requires the use of a window length which is a power of two and does not allow nonstationary behavior of the sinusoids within the window.
In “Refining the digital spectrum”, Circuits and Systems, 1996, by P. David and J. Szczupak, a method is described which allows to estimate the amplitudes and frequencies. This method relies on two spectra of which the second one is delayed in time. In addition the effect of the window is reduced by a matrix inversion which requires a complexity O(K3) for a K×K matrix.
The amplitude estimation methods of the prior art can be categorized in two classes:                Sequential methods compute the parameters for each sinusoid in a sequential manner, i.e. sinusoid by sinusoid. Several methods have been claimed previously:                    1. WO 90/13887 discloses the estimation of the amplitudes by detecting individual peaks in the magnitude spectrum, and performing a parabolic interpolation to refine the frequency and amplitude values.            2. In WO 93/04467 and WO 95/30983 a least mean squares method called analysis-by-synthesis/overlap-add (ABS/OLA) is disclosed for individual sinusoidal components.                        The sequential methods have the advantage that they can be computed very efficiently. However, in case of overlapping frequency responses their result is suboptimal which makes that they cannot be applied when small analysis windows are used. Therefore, the use of large analysis windows is required. However, the definition of the model relies implicitly on the assumption that the amplitudes and frequencies are constant over the analysis window. This assumption is not valid in the case of large analysis windows and results in a poor quality.        Simultaneous methods allow to take into account the overlap between the frequency responses of different sinusoidal components. A method which takes into account the overlap allows to use smaller analysis windows and results in a better quality since the assumption of constant amplitude and frequency is more likely to hold. However, the methods of the prior art known from the literature have a high computational complexity. For instance, the time complexity for the amplitude computation of stationary sinusoids is O(K2N).There is a need for a simultaneous method for analyzing sound signals with a lower computational complexity.        